Course overview

Welcome to Math 593, Graduate Algebra 1! This is an overview of the whole course. You might also be looking for

• A day to day summary of topics and progress.
• A list of assignments and homework policies.

The Euclidean algorithm as it appears in Euclid's Elements, Book VII (source Links to an external site.).

Course meets: Monday, Wednesday, Friday 2-3 PM, 1068 East Hall.

Professor: David E Speyer Links to an external site., speyer@umich.edu, 2844 East Hall.

Office hours: Office hours will be Mondays 11:00 AM - 1:00 PM and Wednesdays 12:00 PM - 1:00 PM, in East Hall 2844.

In addition, our GSI, Saket Shah, will hold office hours 3:00-4:00 PM on Thursdays, in the upper atrium of East Hall.

Topics

This course covers

• Basic constructions with rings and modules
• Euclidean domains, principal ideal domains and unique factorization domains.
• Structure theory of modules and matrices over PID's. Smith normal form.
• Polynomial rings in many variables and Gauss's lemma. Ideally, integrally closed domains.
• Advanced linear algebra topics such as tensor products, exterior algebras, symmetric and alternating bilinear forms, Jordan and rational normal form.

I would describe this as the basic collection of tools for working with rings and modules that any graduate student should know.

Intended audience

The mathematical prerequisites for this course are relatively small: Prior exposure to the definitions of groups, rings, modules and fields. Kernels, cokernels and images of maps of abelian groups, and their basic properties. Abstract linear algebra over an arbitrary field. 

That said, this course is primarily intended for graduate students in mathematics, and will be targeted at such students. Undergraduate students should usually take Math 493. Links to an external site.

Course structure

This class will be taught in an IBL style, meaning that a large portion of the class time will be spent solving problems that develop the theory we are studying. I am indebted to Stephen DeBacker for writing problem sheets to make this possible when he taught the class in Fall 2018; I have extensively modified these problem sheets for the upcoming term. Students are expected to attend class and participate in solving problems, as the class will not work otherwise. Some portion of your grade will be allocated for participation in class work.

In past years, the course was taught entirely in IBL style. Students had a very good experience, but we always wound up very rushed at the end. This year, I plan to assign somewhat fewer IBL problems and do a bit more lecturing, in order to spread out the rushing over the term; however, the class will still primarily be taught in an IBL style.

Climate

 Each of you deserves to learn in an environment where you feel safe and respected.

I want our classroom, the collaborations between my students outside class, and our department as a whole, to be an environment where students feel able to share their ideas, including those which are imperfectly formed, and where we will respectfully help each other develop our understanding. I want to provide a space where questions are very welcome, especially on basic points.

Please ask all questions you have; remember that every question you have is likely a question that many share. Please share your insights and suggestions, partial or complete. Please treat your peers' questions, comments and ideas with respect. 

A welcoming environment includes an environment where each student's disabilities are respected and accommodated. Since this course does not have timed exams, the most common accommodation of extended exam time does not apply, but I am glad to learn about any way that I can make the course more welcoming to you. The office of Services for Students with Disabilities Links to an external site. (734-763-3000 or ssdoffice@umich.edu) provides help for students with disabilities on an institutional level and may be of help. You can also contact me directly. I am also glad to hear about and address barriers to participation which are not disability-related.

Homework

I will assign weekly problem sets, due at 11:59 PM on Fridays through Gradescope Links to an external site.. Here are a list of homework policies and assignments. Some of the problems will include writing up full proofs of results proved in class in IBL sessions.

Grading

A numerical score will be computed as follows: 

• 50% homeworks not including the write ups of class problems, with the lowest two weeks dropped
• 30% write ups of class problems, with the lowest two weeks dropped
• 20% Class participation

 These numerical scores will be converted into letter grades typical of graduate courses.